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Transcendental and algebraic numbers. (Трансцендентные и алгебраические числа) (Transzendente und algebraische Zahlen.) (Russian) Zbl 0048.03303

Moskva: Gosudarstv. Izdat. Tekhn.-Teor. Lit., 224 p. (1952).
This book will be welcomed not only by the Russian student who finds in it an easy introduction to the theory of transcendental numbers, but also by the foreign mathematician. Most of the material in this book has already appeared elsewhere; however, this does not decrease the convenience of having it now available in a very readable form.
The first chapter of the book deals with the Thue-Siegel theorem on the approximation of algebraic numbers by numbers of a fixed algebraic field, and with the \(p\)-adic analogue. The author slightly generalizes the theorem and obtains the same improved exponent as was found independently by Dyson. An application to the class number of imaginary quadratic fields, due to Linnik and the author, concludes the chapter.
Chapter 2 deals with the theorem of Hermite-Lindemann on the transcendency of the exponential function, and with Siegel’s work on the transcendency of the solutions of linear differential equations, in particular the Bessel functions. The proof is based on that in C. L. Siegel’s book [Transcendental numbers. Princeton, N. J.: Princeton University Press (1949; Zbl 0039.04402)].
The third chapter is the most interesting one. It begins with a theorem on integral functions assuming, in a certain set of points, only integral values of not too large heights in a fixed algebraic field. If such a function does not increase too rapidly, it must satisfy a functional equation of the type \(\sum_{h=0}^m A_hf(z+a_h)=0\).
The next subjects considered are the Gelfond-Schneider theorem on the transcendency of \(\alpha^\beta\), and one of Schneider’s theorems on elliptic functions. After this, a measure of irrationality for \(\log \alpha/\log \beta\), where \(\alpha\) and \(\beta\) are algebraic \(p\)-adic numbers, is determined and applied to solve the Diophantine equation \(\alpha^x + \beta^y = \gamma^z\) \((\alpha,\beta,\gamma\) algebraic numbers) effectively in positive integers.
The book concludes with further results on the transcendency of powers and on measures of transcendency.

MSC:

11J81 Transcendence (general theory)
11J82 Measures of irrationality and of transcendence
11-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory
11J17 Approximation by numbers from a fixed field
11J61 Approximation in non-Archimedean valuations
11J68 Approximation to algebraic numbers
11J89 Transcendence theory of elliptic and abelian functions
11J91 Transcendence theory of other special functions

Citations:

Zbl 0039.04402